Considering geological information (breaklines, barriers)
Soil parameters (infiltration rate, soil water content, transmissivity)
Random behavior
Methods:
Geostatistical methods
Considering prior information (e.g. Soil maps)
Topography
not really a random process
Methods:
Exact method to reproduce measurements
See DEM
Oscillation problem of polynomial interpolation
Oscillation problem of polynomial interpolation
Oscillation problem of polynomial interpolation
Oscillation problem of polynomial interpolation
“optimal” degree: number of observation twice the number of coefficients
Be careful at the border (extent the study area!)
Creating a TIN
Predicting value a linear combination of the node values
No jumps at he edges
Jump of the first derivative
RBF in ArcGIS
only by using the Geostatiscal analyst
For all methods except inverse multiquadric, the higher the parameter value, the smoother the map; the opposite is true for inverse multiquadric.
Geostatistical Wizard
Detailed control on all the parameters
Kriging workflow using the Geostastical Analysts
explore the data
Histogram—the distribution of a dataset.
VoronoiMap—stationarity and spatial variability of a dataset.
Normal QQ Plot and General QQPlot—for normality of a dataset and exploration of whether two datasets have the same distribution.
TrendAnalysis—trends in a dataset.
Semivariogram/CovarianceCloud—the spatial dependencies in a dataset.
CrosscovarianceCloud—the covariance between two datasets.
Kriging workflow using the Geostastical Analysts
2. Use the Geostatistical Wizard
select the method
create the empirical semivariogram
Spatial autocorrelation quantifies the assumption that things that are closer are more alike than things that are farther apart. Thus, pairs of locations that are closer (far left on the x-axis of the semivariogram cloud) would have more similar values (low on the y-axis of the semivariogram cloud). As pairs of locations become farther apart (moving to the right on the x-axis of the semivariogram cloud), they should become more dissimilar and have a higher squared difference (move up on the y-axis of the semivariogram cloud).
3. Fit a model to the empirical semivariogram
The empirical semivariogram and covariance provide information on the spatial autocorrelation of datasets. However, they do not provide information for all possible directions and distances. For this reason and to ensure that kriging predictions have positive kriging variances, it is necessary to fit a model (in other words, a continuous function or curve) to the empirical semivariogram/covariance.
The equations for kriging are contained in matrices and vectors that depend on the spatial autocorrelation among the measured sample locations and prediction location. The autocorrelation values come from the semivariogram model. The matrices and vectors determine the kriging weights that are assigned to each measured value in the searching neighborhood.
5. Make a prediction
From the kriging weights for the measured values, you can calculate a prediction for the location with the unknown value.
Summary
Interpolation is not a craft, but an art
the selection of the right tool depends on
the input data (quality, distribution, density, expected spatial behavior)
the experience of the user
use different interpolation methods and compare the results
Photogrammetric processing of stereo aerial photos
Laserscanning (ALS or LIDAR)
Interpolated digitized contour lines from topographic maps (still one of the most important method) drawbacks:
unfavorable distribution of points: dense on contours, gaps in between
generalization on the original map depending on the map scale
improvements:
add singular points for prominent morphological lines and points
use specialised interpolation algorithm, e.g. Topo to Raster in ArcGIS
Topo to Raster (contour interpolation)
is a specialized tool for creating hydrologically correct raster surfaces from vector data of terrain components such as elevation points, contour lines, stream lines, lake polygons, sink points, and study area boundary polygons.
Resolution of Raster DEM for hydrological applications
high resolution low processing time and a lot of storage space needed
what is the appropriate resolution for drainage calculation?
calculation of a time-area diagram to determine a instantaneous unit hydrograph (IUH)
characteristic parameters of the stream network
Channel Length
The distance measured along the main channel from the watershed outlet to the end of the channel
The distance measured along the main channel between two points located 10 and 85% of the distance along the channel from the outlet
Channel Slope
Drainage Density
The drainage density, ratio of the total length of streams within a watershed to the total area of the watershed A high value of the drainage density would indicate a relatively high density of streams and thus a rapid storm response. Values typically ranges from 1 to 4 1/km.
Watershed delineation
Maximum flow length
Algorithm based on D8
start at the outlet cell
look for the source cell with highest accumulation
add 1 (or √2 if it is the diagonal) to the flow length and continue with this cell as new target cell
repeat the previous step as long there are source cells
Watershed characteristics:
drainage area (number of pixel x pixel size)
Watershed or hydrological Length (distance measured along the main channel from the watershed outlet to the basin divide)
Length to the centroid (Lca): the distance measured along the main channel from the basin outlet to the point on the main channel opposite the center of area
Shape Factor Ll = (LLca)0.3 where L is the length of the watershed in km
Circularity ratio Fc = P/(4πA)0.5 Where P and A are the perimeter and area of the watershed, respectively
Circularity ration Rc = A/Ao where A0 is the area of a circle having a perimeter equal to the perimeter of the basin.
Ruggedness Number = basin relief times drainage density
GIS and surface runoff
Spatial aggregated models for flood prediction
input data:
DEM
roughness coefficient for sheet and gully flow
Workflow:
cleaning the DEM (fill sinks)
burn-in topographic stream network
calculation of flow direction map
calculation of slope map
calculation of flow velocities (overland and channel), e.g. using Manning’s equation:
Workflow cont’:
calculation of flow time
v = l / t t = l / v flow length will calculate the distance, with 1/v as weight for each pixel the flow time will be calculated (isochron map)
time-area diagram: histogram of isochron map
IUH: gradient of the time-area diagram
Semi-distributed systems
middle course between distributed models (data requirements!) and aggregated models (generalization)
considering of physical laws as well as simple to use
a large number of models like
COSERO (Continuous semi-distributed runoff model)
SLURP ((Semi-distributed Land Use-based Runoff Processes)
Topmodel (Topography based hydrological model)
TOPMODEL predicts catchment water discharge and spatial soil water saturation pattern based on precipitation and evapotranspiration time series and topographic information. A minimum of four effective catchment parameters need to be estimated to characterize the discharge dynamics of the catchment. The parameters are fitted from the discharge predictions. Neither horizontal or vertical soil parameters need to be supplied. However, to estimate water table or soil moisture content from the saturation deficit requires soil information. A correct estimation of evaporation is critical for model performance. Evaporation is most frequently estimated by using the Penman-Monteith methods.
principles of the model
Surface runoff is computed based on variable saturated areas, subsurface flow using a simple exponential function of water content in the saturated zone. Channel routing and infiltration excess overland flow are considered in the model. The structure of the model with regard to interception and root zone storage compartments is variable, allowing much flexibility to simulated different systems. Time steps should be in the range of an hour to represent surface runoff peaks. The length of the simulation period depends on the availability of precipitation and evapotranspiration input data. The spatial component requires a high quality DEM (digital elevation model) without sinks. remarks: TOPMODEL is also integrated in GRASS GIS version 5
Distributed models
based on physics using energy/momentum equations
high resolution and accuracy required
in one cell homogeneous process assumed
surface runoff
numerical solution of the kinmatic wave equation, e.g.Kineros or lisflood
for uniform flow Chézy or Manning equation where the hydraulic velocity is a function of slope and roughness of the surface
empirical methods like curve numbers
Topographic factors effecting surface runoff can be easily determined using GIS:
slope (implemented)
slope length (not implemented, approximated by distance from stream network
slope form (implemented)
aspect for sun radiation or main wind direction (implemented)
The contribution of GIS
loose coupling, i.e.
data preparation like determination of HRU, interpolation of rainfall data and other model parameter
visualization of results using backdrop maps, hill shading, 3D scenes etc.
Summary
many tools available for physical models as subsurface runoff is 2D!
very important is the DEM as many runoff parameters are based on derivatives of topography
DEM usually has first to be “cleaned”
GIS and Groundwater Modeling
integrated numerical solutions
specialized post-processing tools
calculating analytical solutions based on map algebra
hydrological estimations as map overlays
coupling of GIS and numerical models
Integrated numerical solutions
based on a discretization in space and time of the subsurface water system (finite elements or finite differences)
very costly
orientation of the raster (square parallel to axes versus direction of the main axes of the transmissivity tensor)
e.g. GRASS a FD model directly implemented
Specialized post-processing tools
In the Spatial Analyst of ArcGIS the following tools are implemented:
Darcy Flow and
Darcy Velocity
Particle Track
Porous Puff
DarcyFlow and DarcyVelocity, in conjunction with ParticleTrack and PorousPuff, can be used to perform rudimentary advection-dispersion modeling of constituents in groundwater. This methodology models two-dimensional, vertically mixed, horizontal, and steady state flow, where head is independent of depth.
Darcy Flow Analysis
Darcy's Law states that the Darcy velocity q in a porous medium is calculated from the hydraulic conductivity K and the head gradient (the change in head per unit length in the direction of flow in an isotropic aquifer) as:
q = - K h
where K may be calculated from the transmissivity T and thickness b as K = T / b.
This q, with units of volume/time/area, is also known as the specific discharge, the volumetric flux, or the filtration velocity.
Darcy Flow Analysis
Closely related to this volumetric flux is the aquifer flux U, which is the discharge per unit width of the aquifer (with units of volume/time/length):
U = - T h
This construction assumes that head is independent of depth so that flow is horizontal.
The average fluid velocity within the pores, called the seepage velocity V, is the Darcy velocity divided by the effective porosity of the medium:
V = q/n = (- K h) /n = - (T h) /(bn)
In the Darcy Flow implementation, it is this seepage velocity V that is calculated on a cell-by-cell basis.
(for more information about the implementation see the ArcGIS online help: Explanation of Darcy’s Law)
Darcy Flow Analysis
The purpose of Darcy flow analysis is twofold. First, it is used to check the consistency of groundwater datasets and to generate rasters of groundwater flow vectors. The standard output raster is the groundwater volume balance residual raster, which measures the difference between the flow of water into and out of each cell.
The first step in groundwater flow modeling is to determine the flow velocity and direction at each point in the flow field. Darcy Flow does this and calculates the volume balance within each cell, which should be small in the absence of sources or sinks, such as wells, infiltration, or leakage. A zero volume balance residual indicates a balance between flow in and flow out of the cell. The flow field is assumed to be steady (constant in time).
Darcy Flow Analysis
The differences between Darcy Flow and Darcy Velocity are:
Darcy Flow produces an output volume raster; Darcy Velocity does not.
Darcy Velocity outputs only direction and magnitude rasters as required output; Darcy Flow produces these outputs optionally.
Input:
groundwater head elevation raster.
effective formation porosity raster
saturated thickness raster
formation transmissivity raster
Darcy Flow Analysis
Major drawback consistency of input raster:
However the head elevation raster is obtained, the head must be consistent with the transmissivity raster. That is, the head must reflect the flow through the transmissivity field. It is not sufficient to use values obtained by measurement and testing in the field—the rasterized values must be analyzed for consistency with the aid of a proper porous medium flow program. Consistency implies that the heads would actually be produced by the modeled transmissivity field. Since the true and modeled transmissivity fields often differ in practice, the true and modeled head fields differ as well. Check the heads for consistency by examining the residual raster produced by DarcyFlow. The residual will reflect the consistency of the dataset. Any analysis using DarcyFlow on inconsistent datasets will produce meaningless results.
Particle Track
Tracking a particle through a given velocity field usind a predictor-corrector algorithm
Porous Puff
Solute transport in a porous medium involves two principal mechanisms: advection and hydrodynamic dispersion. Advection describes the passive transport of a solute with the transporting fluid. Dispersion is the mixing of the solute with the fluid by differential movement of the fluids through pore spaces. The Porous Puff function assumes the aquifer is vertically mixed—that is, the concentration is the same throughout a vertical section. So a two-dimensional model can be applied.
For the mathematical background of the implementation see the online help Dispersion modeling with Porous Puff
s = Q /(2π K H) ln(r/R)
where
s lowering of water table (cone of depression)
r distance from the well‘s axis
R distance of influence
Q well discharge
K transmissivity
H thickness
Calculating analytical solutions based on map algebra
if there exist an analytical formula based on two-dimensional input map algebra can be used
example: finding “optimal” location for a new well where “optimal means” lowest sinking of the groundwater table
use Map Algebra to calculate result for each raster cell
Hydrological estimations as map overlays
without solving of the flow or transport equation
map overlay to determine index e.g. for protection zones based on different input maps (vulnerability analysis)
e.g. DRASTIC index for groundwater contamination
Depth to groundwater
Net Recharge
Aquifer media
soil media
Topography
Impact of vadose zone
Conductivitiy
input maps are assigned weight, the summed up weights (map overlay) define the index
[ D ] Depth to water table: Shallow water tables pose a greater chance for the contaminant to reach the groundwater surface as opposed to deep water tables.
[ R ] Recharge (Net): Net recharge is the amount of water per unit area of the soil that percolates to the aquifer. This is the principal vehicle that transports the contaminant to the groundwater. The more the recharge, the greater the chances of the contaminant to be transported to the groundwater table.
[ A ] Aquifer Media: The material of the aquifer determines the mobility of the contaminant through it. An increase in the time of travel of the pollutant through the aquifer results in more attenuation of the contaminant.
[ S ] Soil Media: Soil media is the uppermost portion of the unsaturated / vadose zone characterized by significant biological activity. This along with the aquifer media will determine the amount of percolating water that reaches the groundwater surface. Soils with clays and silts have larger water holding capacity and thus increase the travel time of the contaminant through the root zone.
[ T ] Topography (Slope): The higher the slope, the lower the pollution potential due to higher runoff and erosion rates. These include the pollutants that infiltrate into the soil.
[ I ] Impact of Vadose Zone: The unsaturated zone above the water table is referred to as the vadose zone. The texture of the vadose zone determines how long the contaminant will travel through it. The layer that most restricts the flow of water will be used.
[ C ] Conductivity (Hydraulic): Hydraulic conductivity of the soil media determines the amount of water percolating to the groundwater through the aquifer. For highly permeable soils, the pollutant travel time is decreased within the aquifer.
Coupling of GIS and numerical models
Examples of coupling:
MODFLOW and MFINTER (ArcView 3.x)
UHP-HRU with GUI-UAS (ArcGIS ArcHydro)
MIKE SHE (ArcView)
…
Summary
Groundwater analysis requires 3D, not really supported by GIS
Simplified models which can be reduced to 2D
GIS as pre- and post-processor for numerical models
